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A209750 Triangle of coefficients of polynomials v(n,x) jointly generated with A209749; see the Formula section.

%I A209750 #9 Aug 07 2025 04:35:27
%S A209750 1,2,1,2,4,1,3,6,7,1,3,11,15,11,1,4,14,32,32,16,1,4,21,51,79,61,22,1,
%T A209750 5,25,84,152,174,107,29,1,5,34,118,277,393,352,176,37,1,6,39,172,447,
%U A209750 796,915,666,275,46,1,6,50,225,705,1446,2060,1965,1193,412,56,1
%N A209750 Triangle of coefficients of polynomials v(n,x) jointly generated with A209749; see the Formula section.
%C A209750 For a discussion and guide to related arrays, see A208510.
%F A209750 u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
%F A209750 v(n,x)=u(n-1,x)+x*v(n-1,x)+1,
%F A209750 where u(1,x)=1, v(1,x)=1.
%e A209750 First five rows:
%e A209750   1
%e A209750   2   1
%e A209750   2   4    1
%e A209750   3   6    7    1
%e A209750   3   11   15   11   1
%e A209750 First three polynomials v(n,x): 1, 2 + x, 2 + 4*x + x^2.
%t A209750 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209750 u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209750 v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
%t A209750 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209750 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209750 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209750 TableForm[cu]
%t A209750 Flatten[%]    (* A209749 *)
%t A209750 Table[Expand[v[n, x]], {n, 1, z}]
%t A209750 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209750 TableForm[cv]
%t A209750 Flatten[%]    (* A209750 *)
%Y A209750 Cf. A209649, A208510.
%K A209750 nonn,tabl
%O A209750 1,2
%A A209750 _Clark Kimberling_, Mar 14 2012